For over a century there has been the desire and need to study and predict physical processes such as fluid flow. Understanding the flow past an object (e.g., automobile, airplane wing, etc.) requires an understanding of the turbulence introduced as a result of the relative motion of the fluid and the object. There have been many attempts in the past to gain a better understanding of fluid flow and turbulence.
U.S. Pat. No. 3,787,874, for example, disclosed a method of making boundary layer flow conditions visible by a reactive layer of a colored chemical indicator to the surface of an object exposed to fluid flow. U.S. Pat. No. 4,727,751 disclosed a mechanical sensor for determining cross flow vorticity characteristics. The sensors contained hot-film sensor elements which operate as a constant temperature anemometer circuit to detect heat transfer rate changes.
The direct implementation of vortex methods, however, presents at least two problems that have limited their appeal. First, vortex methods indiscriminately track the dynamics of individual vortical structures as they stretch and fold. As a result, the number of vortices can grow to impractical levels. Second, the computational time associated with the nominally O (Nv2) operations implicit in the Bio Savart law evaluation of velocities due to Nv vortex lemeents, an essential aspect of their formulation and a source of their unique advantages, can grow beyond reasonable levels for engineering use. An additional concern, which pertains to the traditional random vortex method, is the noise introduced by a random walk diffiusion model: unless a very fine coverage of vortex elements is employed, this can easily exceed the levels found in real turbulent flows.
Computer modeling techniques that permit the simulation and analysis of fluid flow using computer modeling is much more attractive because it eliminates the need for physical models and repetitive testing. Some computer simulations attempted to model the fluid flow by covering the entire fluid flow domain with a large mesh or grid. Such grid-based simulations were inefficient and lacking in accuracy to the real physics in fluid flows, particularly those with high Reynolds number turbulence.
With the breakthrough work of Alexandre J. Chorin, University of California in Berkeley, however, a tremendous advantage in understanding and predicting fluid flow was attained using Chorin""s vortex method. According to the vortex method, the vorticity of fluid was used to better understand the basis for fluid flow. In the vortex method a flow field is represented by a collection of convecting and diffusing vortex elements. This method has negligible if any numerical diffusion so that high Reynolds number turbulence effects can be represented with high numerical accuracy and fidelity. The method is grid-free so that it can be easily applied to engineering flows in complex geometries. Moreover, the method is naturally adaptive since the computational elements occupy only the relatively small support of the flow field where the vorticity is significant.
The direct implementation of vortex methods, however, presents at least two problems that have limited their appeal. First, vortex methods indiscriminately track the dynamics of individual vortical structures as they stretch and fold, as a result, the number of vortices can grow to impractical levels. Second, the computational time associated with the nominally O (Nv2) operations implicit in the Bio Savart law evaluation of velocities due to Nv vortex elements, an essential aspect of their formulation and a source of their unique advantages, can grow beyond reasonable levels for engineering use. An additional concern, which pertains to the traditional random vortex method, is the noise introduced by a random walk diffulsion model: unless a very fine coverage of vortex elements is employed, this can easily exceed the levels found in real turbulent flows.
The options for taking account the essential process of vortex stretching over a field of computational elements is a strong function of the particular types of vortices used in the numerical algorithm. The commonly employed vortex blobs, i.e., independently evolving three-dimensional regions of high vorticity concentration, have many desirable properties in the computation of laminar flow solutions, yet are a source of lingering problems when turbulent conditions are encountered. In particular, they tend to be associated with numerical instabilities arising, apparently, when the non-linear coupling between blob strength and stretching causes the former to grow without bound. A common tendency has been to xe2x80x9cswitch offxe2x80x9d the vortex stretching effect to prevent instability, even though this eliminates one of the most fundamental processes of turbulent flow.
New vorticity most often appears in viscous flows at solid boundaries as a natural consequence of the no slip condition, by which fluid in contact with a solid body must move with the same velocity as the solid. Under turbulent conditions, the regions where new vortices appear tend to be relatively thin boundary layers with very high levels of vorticity. The challenge for a vortex method is to efficiently represent such large thin regions with vortex elements while satisfying the no slip condition and generating the proper amount of new vorticity. Typical vortex elements such as blobs and tubes are poorly constructed to represent thin, flat regions, and vortex methods which attempt to use such elements to satisfy no slip tend to be inefficient: large numbers of overlapping elements are needed to counter the inherent three-dimensionality of the velocity field they produce.
In accordance with a preferred embodiment of the invention, a novel apparatus for and method of simulating physical processes such as fluid flow is provided. Fluid flow near a boundary or wall of an object is represented by a collection of vortex sheet layers. The layers are composed of a grid or mesh of one or more geometrically shaped space filling elements. In the preferred embodiment, the space filling elements take on a triangular shape, preferably a triangular prismatic shape. An Eulerian approach is employed for the vortex sheets, where a finite-volume scheme is used on the prismatic grid formed by the vortex sheet layers. A Lagrangian approach is employed for the vortical elements (e.g., vortex tubes or filaments) found in the remainder of the flow domain.
To reduce the computational time, a hairpin removal scheme is employed to reduce the number of vortex filaments, and a Fast Multipole Method (FMM), preferably implemented using parallel processing techniques, reduces the computation of the velocity field.